Robust, Bistable, BMP Dependent Patterning of the Drosophila Blastoderm Embryo: Supporting Online Material(SOM)

Transcription

1 Robust, Bistable, BMP Dependent Patterning of the Drosophila Blastoderm Embryo: Supporting Online Material(SOM) David Umulis 1, Mihaela Serpe 2,3, Michael B. O Connor 2,3, Hans G. Othmer 4 1 Department of Chemical Engineering and Materials Science 2 Department of Genetics, Cell Biology and Development 3 Howard Hughes Medical Institute 4 School of Mathematics and Digital Technology Center University of Minnesota, Minneapolis, MN 4 July 7, 26 1

2 Contents 1 Extracellular patterning model Model assumptions Module I: Heterodimer formation 3 Module II: Extracellular pattern formation 6 4 Module III: Receptor interactions and positive feedback Receptor role in the formation and interpretation of the extracellular gradient Receptor binding (Case 1) Receptor-mediated degradation (Case 2) Derivation of local dynamics for positive feedback of SBP Positive feedback with extracellular decay (Case 3) Positive feedback with endocytosis (Case 4) Decay and endocytosis (Composite of Case 3 & 4) Positive feedback time lag Parameter values 22.1 Numerical methods Transient evolution of patterning components 29 7 Parameter sensitivity and robustness Upstream mechanisms of robustness Sensitivity of receptor interactions Sensitivity to parameters of Module I and II Abnormal model behavior: peak splitting

3 List of Figures 6 Patterning Modules Different receptor models. Symbols are as in Figure Case 1 Results Receptor-mediated endocytosis Case 2 results Schematic representation of SBP action Case 3 results Case 4 results Homotop between Case 3 and Case Effect of a time lag Transient evolution of other patterning components Transient, continued pmad staining in zygotic tkv mutant Amplitude dependence on receptor level Threshold choice Receptor binding sensitivity SBP binding sensitivity Bistability of local dynamics Bistability, continued Sensitivity to extracellular parameters Sensitivity, continued Scale invariance Peak splitting List of Tables 1 Model parameters

4 1 Extracellular patterning model The full model for dorsal surface patterning of the Drosophila blastoderm embryo is composed of three modules or sub-models that, when combined, lead to the sharp, contracting, robust pattern characteristic of this pathway. 1.1 Model assumptions 1. A deterministic description is used for all processes 2. Production of BMP, Sog, and Tsg is continuous 3. The total number of receptors on the surface remains constant 4. Intracellular reactions (Dimerization, signaling, etc.) are rapid and assumed to be at quasi-equilibrium. All non-surface-associated species diffuse 6. Initial amount of unbound receptor is [Rtot ] and Tld is [Tol] 7. Initial amounts of all other species is zero Figure 6: Modules for dorsal surface patterning model with positive feedback of an SBP. 4

5 2 Module I: Heterodimer formation The first module, given in equations 1-3 corresponds to a simplified version of general heterodimer formation of BMPs before secretion into the perivitelline (PV) space [1]. [D] t [W ] t [DW in ] t = φ D (x) K 1 [D][W ] K 2 [D] 2 (1) = φ W (x) K 1 [D][W ] K 3 [W ] 2 (2) = K 1 [D][W ] γ 1 [DW in ] (3) Here, Dpp (D) and Scw (W) bind to form Dpp/Scw (DWin) which is secreted into the PV space. φ D, and φ W are the production rates of Dpp and Scw respectively, and K 1, K 2, and K 3 are the hetero- and homodimer formation rates. In this model, we consider only the heterodimer Dpp/Scw and do not explicitly consider the homodimer counterparts Dpp/Dpp and Scw/Scw. However the results pertaining to extracellular patterning and positive feedback herein are easily extended to the homodimer counterparts, but the robustness will be significantly less for either Dpp or Scw homodimers [1]. main result from (1-3) in [ref. 1] is that the system exhibits insensitivity at the level of Dpp/Scw for 2 fold changes in the level of Scw. u = K 2 /φ W X, v = K 3 /φ W Y, Ω K 1 /2 K 2 K 3, β φ D /φ W gives: The input BMP to the PV space is then: γ 1 [DW in ] = φ wt W 2Ωuv = V in V P V The Defining the dimensionless variables φ Dφ W φ D + φ W and choose Ω=1/2 φ D φ W φ D +φ W. Here V in /V pv = 1*1 3, φ D =1µM*min 1, and φ W =1µM*min 1. The output from this module provides the input to the second module, and there is no feedback to this module from later steps.

6 3 Module II: Extracellular pattern formation Following secretion into the PV space, BMPs interact with extracellular BMP binding proteins Sog and Tsg (see main text: (Introduction) for a brief description). The evolution of the extracellular processes is given by five partial differential equations given at (4-8). The extracellular patterning model used herein is based upon our current understanding of interactions between regulators in the PV space. The extracellular component of this model is similar to those used previously to describe gradient formation during DV patterning [2, 3]. [B] t = D B 2 [B] x 2 + γ 1 V in V P V [DW in ] k 3 [I][B] + k 3 [IB] + λ[t ol][ib] δ B [B] + R([B],[R],[C],...) (4) [S] t [T ] t [I] t [IB] t = D S 2 [S] x 2 + φ S k 2 [S][T ] + k 2 [I] δ S [S] () = D T 2 [T ] x 2 + φ T k 2 [S][T ] + k 2 [I] + λ[t ol][ib] δ T [T ] (6) = D I 2 [I] x 2 + k 2[S][T ] k 2 [I] k 3 [I][B] + k 3 [IB] (7) = D IB 2 [IB] x 2 + k 3 [I][B] k 3 [IB] λ[t ol][ib] (8) Here, extracellular BMP (Dpp/Scw) is denoted (B), Sog (S), Tsg (T), Sog/Tsg (I), and Sog/Tsg/Dpp/Scw (IB). φ T, φ S, and V in V P V DW in are the secretion rates of Tsg, Sog, and BMP into the PV space respectively. Sog and Tsg bind reversibly to form I with constants k 2, and k 2 for the forward and reverse rates. BMP and I reversibly bind with forward and reverse constants k 3, and k 3. R(B,R,C,...) denotes receptor/surface interactions and different cases for R will be analyzed in more detail later. The complex IB is degraded by Tolloid with rate λt ol. Degradation cleaves Sog and releases Tsg and B. The extracellular components in this model undergo non-specific first order degradation/removal. 6

7 4 Module III: Receptor interactions and positive feedback 4.1 Receptor role in the formation and interpretation of the extracellular gradient Formation of the extracellular BMP distribution involves a complex interplay between diffusion and a number of kinetic processes, and the balance between these processes changes along the DV axis. An important factor in localizing Dpp/Scw at the DM is the ability of Sog/Tsg to shuttle Dpp/Scw up the gradient of Dpp/Scw, which has its high point at the DM. Localization of Dpp/Scw at the DM would be enhanced by reducing its diffusion coefficient, and this strategy is used in Model I [2], where the diffusion constant of Dpp/Scw is set to zero. However, some controversy exists surrounding the zero diffusion assumption [3-]. An alternative strategy is to enhance the degradation of Dpp/Scw, for this reduces the amount available for signaling, and since Dpp/Scw is highest at the DM, the signal is highest there. Of course this strategy has the energetic cost of turnover of the signal. A related strategy is used in Model II, in which extracellular BMP is not degraded directly, but rather, degradation is mediated by receptor binding [3]. To better understand these different strategies and how they depend on BMP degradation and receptors, it is helpful to first consider a simple representation of the processes involved, namely, diffusion, signal-independent production and first-order degradation, as given by the equation [B] t = D B 2 [B] x 2 + r ([B], [I], x,...) δb (9) Where (...) includes other model specific components. Scaling time by δ, x by the system length L and assuming B is order one leads to: [B] τ = θ 2 [B] r ([B], [I], x,...) + [B] (1) x 2 δ 7

8 The dimensionless diffusion coefficient θ D B/δ is the square of the ratio of two characteristic L 2 lengths. The first, D B /δ, is a characteristic diffusion length and measures the distance a molecule is spread by diffusion in unit time (measured here in units of the inverse decay rate of BMP), and the second is a characteristic length of the system. If D B /δ is small compared to L, the spread of free BMP is restricted to a small fraction of the domain since it degrades rapidly relative to the rate at which it spreads by diffusion, but if D B /δ is large relative to L, diffusion dominates and the signal spreads rapidly. Thus localization of the signal can be achieved either by having a small diffusion coefficient or a large degradation rate. In our model with linear degradation (no positive feedback) the diffusion length is approximately 6µ, and since the circumference is µ, this suggests that Dpp is moderately localized, which contributes to the sharp spatial gradient. Next we examine the effect that receptors have on localization of BMPs and the receptor interpretation of BMP signal. In Case 1 we consider simple receptor binding dynamics, while in Case 2 we look at receptor-mediated degradation. These cases have no feedback. In Case 3, we study how positive feedback of a membrane-bound binding protein affects the interpretation of the BMP gradient with extracellular degradation of BMP. And lastly, in Case 4 we look at the model of the main text with positive feedback and endocytosismediated degradation of the BMP (Figure 7). 4.2 Receptor binding (Case 1) The simplest interaction arises when BMP binds to the receptor at a rate k on and is released at the rate k off, and this leads to the following equations, wherein BR is the concentration of BMP-occupied receptor and B is the concentration of extracellular BMP. [B] t [BR] t = D B 2 [B] x 2 k on [B] (R tot [BR]) + k off [BR] + r ([B], [I], x,...) δ[b] (11) = k on [B] (R tot [BR]) k off [BR] (12) 8

9 Figure 7: Different receptor models. Symbols are as in Figure 6. Since receptor binding and release are fast relative to diffusion and other extracellular processes, we can assume a quasi-steady state for the BR binding steps, which yields [BR] = [B] [Rtot ]/([B] + kof f /kon ). Thus the level of BR is proportional to the level of extracellular BMP for [B] << kof f /kon, and saturates as B increases. When the extracellular concentration varies in time the receptors act as a capacitance to slow changes. However, at equilibrium the contribution is zero and therefore does not affect the steady-state distribution in space. This simple readout of the extracellular morphogen gradient can produce BR profiles that correspond generally well with pmad staining in the Drosophila embryo 8. Because the level of BR directly reflects the extracellular concentration of BMP, robustness present in the mechanism for establishment of the spatial profile feeds directly into BR. 4.3 Receptor-mediated degradation (Case 2) Recently, a model that incorporated receptor-mediated degradation of BMP was developed (Model II)[3]. The governing equations are: 9

10 BMP (nm) min 3 min 4 min 6 min s.s. BMP-Receptor (nm) min 3 min 4 min 6 min s.s Figure 8: Transient evolution of BMP and BMP-Receptor for Case 1. The position corresponds to the dorsal midline. [B] t [BR] t = D B 2 [B] x 2 k on [B] (R tot [BR]) + k off [BR] + r ([B], [I], x,...) (13) = k on [B] (R tot [BR]) (k off + k deg ) [BR] (14) Here, R tot is the total amount of surface-localized receptors and k deg is the internalization rate of the BR complex. This equation is based on two features: very rapid intracellular processes, and as shown next, free and bound receptors internalize/degrade at the same rate. If we denote the above species as R s, R i, R s L, R i L where R is receptor, s is cell surface, i is internal and L denotes ligand, the following set of ordinary differential equations can be derived. 1

11 d[r S ] dt d[lr S ] dt d[lr i ] dt d[r i ] dt = k on [L] [R S ] + k off [LR S ] k i1 [R S ] + γ [R i ] (1) = k on [L] [R S ] k off [LR S ] k i2 [LR S ] (16) = k i2 [LR S ] λ L [LR i ] (17) = λ L [LR i ] + φ R γ [R i ] + k i1 [R S ] λ R [R i ] (18) Also, φ R is the production of receptor, k i1 and k i2 are the internalization rate for free and Figure 9: Receptor-mediated endocytosis. bound receptors, and λ L and λ R are the ligand and receptor degradation rates respectively. If a quasi-steady-state for the internal processes is assumed, and if we impose a conservation condition on the level of receptors on the surface, then (1),(16)= [LR i ] = k i2 λ L [LR S ] (17)+(18) d[r S] + d[lr S] = d[r ] dt dt dt = = k i1 [R S ] k i2 [LR S ] + γ[r i ]. Now sub (17) into (18) and arrive at k i2 [LR S ] k i1 [R S ] + γ[r i ] = φ R λ R [R i ] and [R i ] = φ R λ R. This gives k i1 [R S ] + k i2 [LR S ] = γφ R λ R. Now = k i1 ([R ] [LR S ]) k i2 [LR S ] + γ φ R d[r λ R k ] i1 = = dt (k i1 k i2 ) d[lr S] dt and k i1 = k i2. Thus, receptor-mediated degradation of the form used in Model II implies that free and bound receptors internalize at the same rate. This has been shown in some contexts experimentally [6] and this result will be used later for derivation of the positive feedback 11

12 mechanism. In this scheme BMP is removed from the extracellular space only when bound to receptor, and some simple assumptions make the role of receptor-mediated degradation more transparent. To identify an appropriate kinetic scale, note that there are two dominant kinetic processes that contribute to the localization of BMP: inhibitor binding/cleavage and receptor binding and degradation. Since the maximum signal range depends on the slowest kinetic process, we focus on the slower receptor internalization and degradation rather than the fast extracellular inhibitor binding. Furthermore, the BR off-rate used was sec 1 in [3] which is very small relative to many of the other kinetic rates (e.g. 1 fold smaller than the degradation rate) and can be assumed to be zero. Thus, the effective receptormediated removal rate of BMP from the PV space depends on the balance of receptors and the equation becomes: [B] t = D 2 [B] }{{ x 2 } Diffusion f ([BR]) [B] + k }{{} off [BR] }{{} Rec.Rem. ϑ() + r([b], [I], x,...) }{{} Binding,... (19) The characteristic degradation parameter δ=f(br) depends upon the level of BMP bound to receptor. Due to the fast binding steps coupled with the slower release and degradation of BMP, the quasi steady-state analysis cannot be invoked a priori so we first analyze the transients for the level of BR. Suppose B varies slowly on the binding time scale variable so that receptors perceive it as constant, then: [BR] (t) α β (1 exp [ βt]), α = k on[r ] [B], β = k on [B] + k off + k deg The approximate time for BR to reach 8% of its equilibrium value for BMP near the midline is ln (.2)/ β 1. min or 2.3 min to reach 9% of it s equilibrium value. For lower levels of BMP, the time to equilibrate is slower, taking on the order of minutes. Since the level of BMP is increasing in the PV space during patterning in this model [3], the level of BR lies somewhere between zero and the steady-state level. For early transient patterning, before receptors equilibrate, the 1 st order removal of BMP from the extracellular space (δ) 12

13 is δ k on [R ]= 1.2sec 1. At steady-state the removal rate of BMP from the PV space is δ (k deg k on [R ])/(k on [B] + k off + k deg ) =.4sec 1. These values for the effective 1 st order removal rates give diffusion lengths D L /k between 8µ and 4µ. Since the circumference of the embryo is µ, this means that the BMP localized by other mechanisms will travel only a very short distance before being bound by receptor and degraded, thus localizing it in the presence of diffusion. The analysis of robustness for this receptor interaction is analyzed in detail in Mizutani et al. [3] and in section 7 herein. BMP (nm) min 3 min 4 min 6 min s.s. BMP-Receptor (nm) min 3 min 4 min 6 min s.s Figure 1: Transient evolution of model with receptor mediated endocytosis of BMP. 4.4 Derivation of local dynamics for positive feedback of SBP In this section we provide further details on two cases that may lead to the positive feedback identified by Wang and Ferguson []. A candidate molecule of BMP receptor activation is a membrane bound BMP binding protein. Figure 11 shows the full proposed mechanism for receptor activation and signaling. In Case 3, the SBP is degraded in all forms at a rate δ D, but receptor is not (δ E = ). BMP-bound SBPs release BMP upon degradation and as a result, positive feedback does not influence the extracellular BMP distribution at steady-state. This mechanism biologically resembles secretion of a molecule that binds extracellular components that restrict its 13

14 movement (perhaps by binding to heparin-sulfate proteoglycans) but is not specifically endocytosed. Case 4 involves endocytosis of BMP when bound to SBPs (or receptors) associated with the membrane which internalize at the rate δ E before undergoing intracellular degradation and/or recycling. For simplicity we assume that BMP decays only in the unbound state, thus the total rate of removal of BMP is the sum of the extracellular decay and the surface-associated endocytosis. The difference between these cases stems principally from the fact that in Case 3 BMP is not lost as a result of binding, while Case 4 leads to a loss of BMP via endocytosis. The latter couples the steady-state distribution of SBP and BMP in the PV space. The simplified mechanism presented in the main text can easily be derived by making a few assumptions. First, the level of receptor complex composed of type I and type II receptors forms by binding of BMP to one type of receptor, which facilitates the recruitment of the other receptor to form an active signaling complex. Once the complex is formed, it starts phosphorylating Mad. If Mad is not limiting, the production of pmad is proportional to the number of bound receptors. Once phosphorylated, pmad binds to the co-smad Medea before translocating to the nucleus to control target genes. Since the binding of pmad to Medea is an example of a dimerization step of the type analyzed in detail previously [1], two cases are possible. In one case, the level of Medea exceeds the level of pmad and the level of pmad bound to Medea is directly proportional to the level of pmad in the system. In the case with limiting Medea, the formation of the complex saturates as the level of pmad increases and the difference between bound states of receptors would be dampened out. For the analysis here, we only look at the case in which Medea is in excess and the level of the pmad/medea complex is directly proportional to the level of ligand bound receptors (i.e. pmad/medea α*activereceptor). Gene expression is modeled using a second order Hill function similarly used in other systems with positive feedback. If a constant level of surface receptors for the case with the positive feedback of the BMP binding protein, we obtain a similar expression and requirement for the internalization rate of the various components as 14

15 Figure 11: Schematic representation of SBP action. derived earlier for Case 2. Here the condition that must be satisfied is: = (ki1 ki2 ) d[brs ] d[cbrs ] + (ki1 ki3 ) dt dt where C denotes the SBP, RS is the surface localized receptors, and B is the BMP. This condition is easily satisfied by setting all the internalization rates equal. For simplicity, we set the internalization rate for the free and BMP-bound surface binding proteins equal to the receptor internalization rate for the computations. The active signaling complex for BMPs is composed of both type I and type II BMP receptors. The equations for the full patterning model are: 1

16 d[c] dt d[bc] dt d[bcr] dt d[br] dt = Λ [BRP ]ν K ν h1 + [BRP ]ν k 4[B][C] + k 4 [BC] k 7 [BR][C] + k 7 [BCR] (δ D + δ E ) [C] (2) = k 4 [B][C] k 4 [BC] k 6 [BC][R] + k 6 [BCR] (δ D + δ E ) [BC] (21) = k 6 [BC][R] + k 7 [BR][C] k 6 [BCR] k 7 [BCR] (δ D + δ E ) [BCR] (22) = k [B][R] k [BR] + (k 7 + δ D ) [BCR] k 7 [BR][C] k 8 [BR][P ] + k 8 [BRP ] δ E [BR] (23) d[brp ] dt = k 8 [P ][BR] (k 8 + δ E ) [BRP ] (24) [R tot ] = [R] + [BR] + [BCR] + [BRP ], [P tot ] = [P ] + [BRP ] (2) where, B denotes BMP, R denotes the type I receptor and P denotes the type II receptor. At steady-state, the level of BRP is [BRP ] = which for BR ([BR] << (k 8 + δ E )/k 8 ), [P tot ][BR] [BR] + (k 8 + δ E ) / k 8 (26) BRP (k 8 /(k 8 + δ E ))P tot BR = K eq BR (27) In the positive feedback model this primarily affects the level of free receptor that can participate in binding reactions with other components. For simplicity, assuming the level of BRP is small relative to R, BR, and BCR, the following equations were used: 16

17 d[c] dt = Λ[BR]ν K ν h + [BR]ν k 4[B][C] + k 4 [BC] k 7 [BR][C] + k 7 [BCR] (δ D + δ E ) [C] (28) d[bc] dt d[bcr] dt d[br] dt = k 4 [B][C] k 4 [BC] k 6 [BC][R] + k 6 [BCR] (δ D + δ E ) [BC] (29) = k 6 [BC][R] + k 7 [BR][C] k 6 [BCR] k 7 [BCR] (δ D + δ E ) [BCR] (3) = k [B][R] k [BR] + (k 7 + δ D ) [BCR] k 7 [BR][C] δ E [BR] (31) [R tot ] = [R] + [BR] + [BCR] (32) and in terms of K eq, K ν h = Kν h1 /(K eq) ν. 4. Positive feedback with extracellular decay (Case 3) In this case, unbound extracellular BMP undergoes first-order degradation at a rate δ B = 1 min 1, and the SBPs decay at a rate δ D =.3 min 1 and δ E =. At steady-state the positive feedback and receptor interactions sum to zero and therefore do not affect the shape of the extracellular morphogen gradient. Plotting the point wise data for extracellular BMP vs. the level of BR on the equilibrium diagram for the local dynamics shows that this mechanism leads to a threshold response to a time-independent distribution of BMP; some cells lie near the lower branch while others lie near the upper branch, and the transition occurs between two adjacent cells (Figure12 a). The threshold level of BMP corresponds to the limit point of the equilibrium curve: cells that detect BMP above that level eventually adopt a high signaling cell fate, while cells below that point adopt a low signaling fate. With this mechanism, two very different extracellular BMP profiles can lead to approximately the same profile of BRs. To demonstrate this, we reduced the binding affinity of Dpp/Scw to Sog/Tsg and lowered the decay rate so the extracellular gradient is broad (dashed line, Figure 12 c). In Figures 12 b and 12 c, the extracellular gradients have a very different 17

18 shape, but the level of BRs is very similar, differing primarily in amplitude near the midline. Since the extracellular ligand distribution is determined independently of positive feedback interactions, the robustness results from earlier models translate directly into the level of BR and thus to signaling at steady-state. Figure 12: Case 3, no endocytic degradation. (a-c) Positive feedback leads to spatial bistability. (a) The steady-state response, as measured by the level of BR as a function of extracellular BMP, treated as a parameter. For low BMP, BR interactions closely follow the binding curve. When the level of extracellular BMP exceeds the lower limit point (purple star), the lower branch disappears and the level of BR approaches the upper branch. Red dots are the level of BMP and BR corresponding to cell positions in the PV space (b). (b-c) Decay mediated removal of BMP and binding protein leads to a threshold response at steady-state. (b) BMP (dashed line; right axis) and the level of BR (solid line; left axis) vs. position at steady-state. (c) Same as in (b) except that the extracellular BMP is less tightly localized and higher throughout by setting δb = min 1 and k3 =1.71 nm 1 min 1. The level of BR output in (b-c) is very similar in width and varies slightly in amplitude. 4.6 Positive feedback with endocytosis (Case 4) Positive feedback acts at two levels to control the transient evolution of patterning. First, it enhances the binding of BMP to the receptor by increasing the local concentration of BMP that has access to receptors. Second, since we hypothesize that SBP/BMP complexes internalize, this creates a transient sink for BMP binding. Effectively, this gives a diffusion length for the system that changes in time. Initially, receptor mediated endocytosis (as in Case 2 above) limits the spread of free BMP. Once SBP is secreted, additional BMP binds to the SBP in addition to the receptors and removed from the PV space. This produces a transiently decreasing diffusion length, which leads to a gradient contraction. Other mech18

19 anisms that also lead to a contracting gradient are time-dependent BMP production [2], time-dependent Sog production or a transient inhibition of the Tld protease (unpublished data). BMP (nm) min 3 min 4 min 6 min s.s. BMP-Receptor (nm) min 3 min 4 min 6 min s.s Position ( ) Position ( ) Figure 13: Transient evolution of model with positive feedback coupled with endocytosis of receptors and SBP (Case 4). Here, R tot =32nM (see section 7 for explanation). 4.7 Decay and endocytosis (Composite of Case 3 & 4). Here we expand the discussion of Cases 3 & 4 from the main text and analyze the transient evolution of the level of BR for intermediate systems in which both processes contribute (with varying degree) to the full patterning process. To do this, we have to balance the total degradation so that the number of cells that reach the upper branch (-7) corresponds to the half-width of high intensity pmad signaling. First, we require that the sum of the degradation rates for decay and endocytosis of the binding protein remain constant (i.e. δ D + δ E = δ tot =.3 min 1 ) so that the level of SBP is consistent. Furthermore, since the protein decay process releases BMP while the endocytosis process removes BMP from the system, we scale the first order decay rate of extracellular BMP by the total (i.e. δ Beffective = (δ D /δ tot )*δ B ) and then the level of extracellular BMP degradation interpolates between the maximum δ B in the decay only case and zero in the endocytosis only case. To understand how the system responds to the intermediate ratios of the different removal processes, we plotted the transient 19

20 evolution of the level of BMP and BR above each cell as described in the main text for Figure 3 c. Figure 14 shows the results for four different ratios of the decay processes. Figure 14 a shows the case with only extracellular decay processes and the transient approach to the upper stable branch for the threshold mechanism (Case 3). The process is very slow because extracellular BMP levels are relatively low even though they exceed the limit point. Since the levels are low, the production of the binding protein in the positive feedback is slow and the evolution of BR to the upper stable branch is slow (much slower than embryonic patterning observed in vivo). Increasing the relative ratio of endocytosis to decay increases the rate at which cells approach the upper branch. Furthermore, this process feeds back into the extracellular patterning system, causing some cells that initially exceed the lower limit point to transiently return to the lower stable solution branch, since other competing cells begin removing the BMP from the extracellular space at a rapid rate. This leads to an initial broad distribution followed by a contraction. While the patterning time is slightly faster here, the positive feedback is still too slow to affect the pmad distribution and embryonic patterning. By further reducing the proportion of BMP removed by extracellular decay, the features previously noted become more pronounced. The fastest transient evolution with positive feedback occurs for Case 4 in which BMP is only removed by endocytosis when bound to either receptors or the SBP. Also, Case 3 does not lead to a contraction at steady-state since all cells that sense an extracellular level of BMP that is greater than the lower limit point eventually reach the upper stable branch (Figure 14 a). As BMP is removed via endocytosis as in Case 4, some cells return to the lower equilibrium branch due to the rapid uptake of extracellular BMP by competing cells (Figure 14 b-d). 2

21 a BMP-Receptor (nm) c BMP-Receptor (nm) min Extracellular BMP (nm) Increasing time δ D 1 δ + δ = D δ E D δd + δ = E 6 min. Increasing time 1 3 b BMP-Receptor (nm) d BMP-Receptor (nm) Extracellular BMP (nm) min. δ Increasing time D δd + δ = E 6 min. 2 3 δ D δ + δ = D E Increasing time Extracellular BMP (nm) Extracellular BMP (nm) Figure 14: Homotop between Case 3 and Case Positive feedback time lag Up to this point it was assumed that the transcriptional lag time for positive feedback is zero. However, it may be the case that signaling, gene control, and production of the SBP may have a lag time that may change both the transient evolution and steady-state distributions of patterning. To account for the positive feedback lag, changes were made in the equation 21

22 for SBP by introducing the term (t τ). d[c] dt = Λ[BR]ν K h1 + [BR] ν (t τ) k 4[B][C] + k 4 [BC] k 7 [BR][C] + k 7 [BCR] (δ D + δ E ) [C] (33) Here τ is a generic lag time that is specified as a parameter and this is commonly used in transient models with positive feedback. To compute transient patterns with the time lag, we programmed a delay differential equation (DDE) solver in Matlab and computed evolutions for long times. Figure 1 shows the transient trajectories of dorsal cells for τ =, and 1 min. As the time lag associated with positive feedback is increased, there is significantly higher extracellular BMP during the transient evolution (compare Figure 1 a with b and c) along with a delay in the transient contraction. As patterning commences, the contraction is more pronounced for greater time-lags, however the time-scale for patterning becomes too long to be biologically meaningful. This suggests that either the positive feedback lag during patterning is small, or that the time-scale for the modeled system is incorrect for the specified set of parameters. The time-scale limiting parameter for the current model is the endocytosis rate δ E which is based on earlier studies [3]. Our analysis suggests that when this constraint is relaxed, and faster endocytosis rates are used, the transient evolution can occur in the required time-scale even with a transcription time-lag (unpublished data). Parameter values The concentrations of the BMPs, inhibitors, and proteases are not precisely known. Parameter values were estimated based on in vitro experimental data, biacore binding data, and other sources. Inhibitor binding parameters(k 2,k 2, k 3,k 3 ). The binding parameters used are in the biologically reasonable range and based on previous studies where available. Furthermore, as 22

23 a. b. BMP-Receptor (nm) 4 τ = min. 3 6 min BMP-Receptor (nm) min. 3 min. 4 min. 6 min. s.s. c. d. BMP-Receptor (nm) τ = min. 6 min Extracellular BMP (nm) e. f. BMP-Receptor (nm) Extracellular BMP (nm) τ = 1 min. 6 min. BMP-Receptor (nm) BMP-Receptor (nm) min. 3 min. 4 min. 6 min. s.s min. 3 min. 4 min. 6 min. s.s Extracellular BMP (nm) Figure 1: Effect of a transcription/production lag on patterning. Results are shown at for min. (a,b), min. (c,d) and 1 min. lag times (e,f). 23

24 shown later, the dynamics of the model do not depend on the specific choices of parameters and the model is insensitive to a large number of parameters when varied independently over 3 orders of magnitude (see section 7.2). The Sog and Tsg forward binding rate and the maximum value for B and I forward binding were approximated from previous work on extracellular BMP patterning and are given in Table 1 [3]. Receptor levels and SBP Production. The concentration of receptors that participate in transmitting the BMP signal from the PV space to the nuclei is unknown. The average number of receptors in 12 different cell types is approximately receptors with a range of to [7]. For example, there are 1 3 activin receptors per cell in the Xenopus blastula [8]. The area that each responding nucleus in the syncytium has exposed to the PV space is approximately a µ X µ patch of the PV membrane or approximately 1/6 th its total surface area. This gives a range of receptor numbers in the PV space of 1 to receptors per nucleus. The volume of the PV space per nucleus is 12.µ 3 and the receptor concentration can be computed by R tot = ρ/(v N A ) where ρ is the number of receptors, V is the volume, and N A is Avogadro s number ( ). This gives a range of receptor concentration between 39 and nm. The maximal production rate of the SBP (Λ) was set at 18 molecules/cell/minute which corresponds to 24nM*min 1 which is in the same order of magnitude for receptor production rates used in similar studies [7, 9]. BMP and inhibitor levels. Using cell-based signaling assays, Shimmi and O Connor demonstrated that Drosophila line S2 cells respond to Dpp concentrations <1nM [1]. In fact, pmad signaling observed in S2 cells changes from basal levels when Dpp concentration is 1 11 M to maximal expression when Dpp concentration is 1 9 M. While this information suggests signaling can occur at very low concentrations of Dpp, this does not guarantee concentrations in vivo are in the nm range. Nevertheless, the cell-based signaling assay provides an estimate for the concentrations of BMP species. Second, it was shown that in vitro cleavage reactions exhibit the same dose dependency 24

25 observed in the cell-based signaling assays. When Sog was incubated with Tld and Dpp, strong proteolytic processing occurs at Dpp concentrations of 1nM. Furthermore, the in vitro reaction cleaves fragments off Sog after 1 minutes when incubated with M Tsg and Dpp. Patterning time for the amnioserosa takes about 3-4 min. in vivo which is comparable with the time cleavage takes in vitro. These experiments provide evidence that suggests concentrations for BMPs and Tsg in vivo may be in the 1-1 nm range. Sog production (φ S ) was set to 4nM*min 1, while Tsg production(φ T ) is set to 36nM*min 1. φ D and φ W are given in section 2. The level of extracellular Tld is fixed at 6nM. From section 2, The input BMP to the PV space is: V in V P V φ D φ W φ D +φ W. Here V in /V pv = 1*1 3, φ D =1µM*min 1, and φ W =1µM*min 1. On a concentration basis, this gives heterodimer BMP inputs into the PV space on the order of 1nM*min-1. In Case 3, the 1 st order degradation/internalization for the BMP (δ B )is set at 1.min 1. This is fairly rapid and gives a diffusion length of 66nM. Diffusion coefficients. The simplest correlation based on the molecular weight, temperature, and solution viscosity provides a starting point for estimation of the diffusion of the species within the PV space. The equation relating temperature (T), viscosity (cp) and molecular weight (g mol 1 ) is shown below [11]. D = ( ) T ηm 1/3 cm 2 sec 1 (34) Using this equation, the diffusion constants for the species involved were calculated assuming 1 cp for the viscosity of water and 298 o K for T. The MW and calculated diffusion rate for the green fluorescent protein (GFP) is also calculated using the same equation. GFP, a 238 amino acid, 27 kd protein has an experimentally-determined diffusion coefficient of 8 µ 2 sec 1 in water. Thus for GFP, the equation yields an error of approximately 3.%. Receptor binding parameters. With low levels of morphogen in the PV space, it would seem that the ligand binding rate to receptor would have to be very high to ensure occupancy sufficient to signal. However, previous analysis suggested that in order for a suitable 2

26 gradient to form by diffusion, the forward step in binding has to be relatively low to ensure patterning is not too steep and acts over a range to be biologically useful [12]. In Drosophila embryonic patterning, the morphogen gradient begins broad and low and refines to a narrow stripe near the dorsal midline, which is the opposite of what occurs in classical morphogen gradient models where the gradient originates at secreting cells and extends through the developmental field by diffusion or other processes. In this context a rapid forward rate would limit the spread of BMP, but this would also have the negative effect of binding BMP and degrading it before the extracellular transport machinery effectively translocated the BMP from its region of expression to the dorsal midline [3]. In fact, the binding on-rate of BMP2 (Dpp like) to BMP Type I Receptors (BRIA) (Tkv like) is relatively slow with a k on between nm 1 min 1 and nm 1 min 1 and the value used herein for the forward rate binding of the heterodimer Dpp/Scw (k ) was nm 1 min 1 [13, 14]. Typical dissociation constants (K d ) for BMP-2 with receptors is relatively high: K d 1 nm (Biacore) for the immobilized BMPR-IA ectodomains or nm for the immobilized BMP-2 [14]. Another study found that BMP-2 binding with Tkv receptor also has a higher affinity with immobilized receptor (3.6 nm) versus ( 2 nm) for immobilized BMP-2 [13]. This suggests that for BMP-2 homodimer binding, the expected off-rate (if an on-rate of nm 1 min 1 is assumed) is between min 1 and 4.8min 1. The discrepancy in K d values for the two different immobilized factors may be caused by two immobilized receptors binding to a single ligand, which may be the more biologically relevant case. Since the ligand involved in Drosophila patterning is likely a heterodimer of Dpp/Scw [1], binding rates of the Scw like BMP-7 were estimated. Receptor binding of BMP-7 (Scw like) to type I BMP receptors is significantly lower with K d values for immobilized receptor that are either medium K d = 2 1nM or low K d = 1 1nM [14]. However, at this time it is unclear if the low K d values is caused by a slower on-rate or a faster off rate. In the modeling of receptor interactions, the details of oligomerization and heterodimer formation were not considered. For additional analysis of heterodimer formation see Shimmi et al. [1] or Vilar et al. [1]. A relatively fast off rate (k )of 4min 1 which gives a K d 166nM was used for the calculations in the main text. See section 7 on parameter sensitivity for additional 26

27 discussion. Surface bound BMP binding protein (SBP) binding parameters. Without the identity of the molecule in the embryonic context that leads to accumulation of BMP at the dorsal surface we can only speculate on appropriate parameter values or even a range of parameter values appropriate for modeling. In the analogous patterning stage during zebrafish gastrulation [16] and during BMP patterning of the pupal wing (our unpublished data) Crossveinless-2 (Cv-2) is the positively regulated component required to potentiate BMP signaling. The binding affinity of BMP-2 for WT Cv-2 is K d 1.4nM with an onrate of nm 1 min 1 and an off-rate of min 1 [16]. For BMP-7 the K d 3.nM with an on-rate of nm 1 min 1 and an off-rate of min 1. The on-rate for both BMP-2 and BMP-7 is about 6-8 fold higher for Cv-2 than for receptor which would lead to accumulation of BMP in regions with high levels of Cv-2. An on-rate (k 4 ) of 1nM 1 min 1 and an off-rate (k 4 ) of 2 min 1 with a K d = 2nM were used in the absence of the aforementioned data regarding Cv-2, however if the above rates are used for Cv-2, there is little effect on the transient evolution or steady-state properties of the BMP-receptor distribution (our unpublished data). The rates for the transfer of BMP between SBP and receptors on the surface are largely unknown at this time and rates were chosen in the biologically feasible range for k 6, k 6, k 7, and k 7. The model behavior is largely insensitive to these choices as shown in section 7..1 Numerical methods In view of assumption 3, the steady-state versions of equations 1-3 were solved using the results for general heterodimer formation [1]. Equations 4-8 were solved for various receptor models using the finite difference method on an axial cross-section halfway between the anterior and posterior poles. The equations were solved on half the circumference of the cross-section by imposing symmetry conditions at the dorsal and ventral midline. We used node points in the half-width, which is approximately the number of prospective cells. In this way, we treat each node point as a cell lining the PV-space and are able to track the 27

28 Name Description Value Units Figure Ω Dimer formation. Dimensionless γ 1 Secretion Rate arbitrary sec 1 Inhibitor Binding k 2 Binding Sog/Tsg 3. * 1 1 nm 1 min 1 2 k 2 Reverse 1.8 * 1 min 1 2 k 3 Binding I to B.7 * 1 nm 1 min 1 26 k 3 Reverse 1.8 * 1 1 min 1 26 Receptor/SBP Binding k 4 Binding B to C 1. * 1 nm 1 min 1 22,23 k 4 Reverse 2. * 1 min 1 22,23 k Binding B to R 2.4 * 1 2 nm 1 min 1 21,23 k Reverse 4. * 1 min 1 21,23 k 6 Binding BC to R. * 1 1 nm 1 min 1 22,23 k 6 Reverse 1. * 1 1 min 1 22,23 k 7 Binding BR to C 1.3 * 1 1 nm 1 min 1 22,24 k 7 Reverse 1. * 1 1 min 1 22,24 Diffusion D S Diffusion, Sog µ 2 sec 1 27 D T Diffusion, Tsg 66 µ 2 sec 1 27 D B Diffusion, BMP 73 µ 2 sec 1 27 D IB Diffusion, I/BMP 42 µ 2 sec 1 27 D I Diffusion I 4 µ 2 sec 1 27 Production, etc. φ S Sog secretion 4 nm*min 1 φ T Tsg secretion 36 nm*min 1 φ D Dpp production 1 µm*min 1 φ W Scw production 1 µm*min 1 V in /V pv Volume ratio 1. * 1 3 Dimensionless Tld Tld levels 6 nm R tot Total Receptor Level 16, 32 nm Degradation λ Tld process rate * 1 nm 1 min 1 δ Sog Sog Degrade 1. * 1 1 min 1 δ T sg Tsg Degrade. * 1 2 min 1 δ B B Degrade 1. * 1 min 1 δ tot, (δ E ) Membrane Prot. Degrade 3. * 1 2 min 1 SBP production Λ Bind. Prot. Production 24 nm*min 1 24 K H Hill parameter nm 24 ν Cooperativity/Gene Param. 2 Dimensionless 24 Table 1: Model parameters. The figure column lists the figure(s) where parameters underwent sensitivity analysis. denotes R tot used for Case 4 in section 7.2. and Figure 2 in main text 28

29 transient behavior of individual cells on standard bi-stability diagrams. Bifurcation diagrams corresponding to the local dynamics of the positive feedback loop were calculated using the software Matcont written for Matlab [17]. 6 Transient evolution of patterning components The primary focus of the analysis and article is on the distribution of BMP bound to receptor. However, BR depends upon the distribution of many other patterning components which are given in Figures 16 and Parameter sensitivity and robustness 7.1 Upstream mechanisms of robustness. It is found experimentally that D/V patterning is surprisingly robust with respect to reductions in the levels of many of the components. For instance, the width of the pmad expression domain is insensitive to reductions in the level of extracellular components Tsg, Scw, and Tld [2] while it shows greater sensitivity to reductions in Sog [3]. Patterning is highly sensitive to changes in the level of Dpp. Earlier analysis of extracellular BMP patterning (Module II) demonstrated that for certain sets of parameters, the computational system exhibits robustness or insensitivity to 2-fold changes in the levels of Sog, Scw, and Tld at steady-state [2]. The conditions for steady-state robustness are (1) BMP dependent processing of Sog by Tld; (2) free BMP molecules do not diffuse independently; (3) BMPs bind irreversibly to receptors; (4) Sog can remove BMP from receptors, and () Dpp and Scw patterning is decoupled by the Sog/Tsg complex. While condition is not met since patterning is likely mediated by a heterodimer Dpp/Scw [1] conditions (1-4) should be considered separately. The net effect of conditions (1-4) is to strongly localize BMP in the absence of Sog binding. That BMPs are localized in the absence of the inhibitor Sog has become controversial since new findings suggest that Dpp can act over 1-2 cell diameters and Scw can act over even 29

30 a. b. Concentration (nm) min. 3 min. 4 min. 6 min. s.s c. d. Concentration (nm) e. f. Concentration (nm) Sog Sog-Tsg 1 min. 3 min. 4 min. 6 min. s.s Sog/Tsg/BMP 1 min. 3 min. 4 min. 6 min. s.s Concentration (nm) Concentration (nm) Concentration (nm) Tsg 1 min. 3 min. 4 min. 6 min. s.s Dpp-Scw 1 min. 3 min. 4 min. 6 min. s.s SBP/BMP 1 min. 3 min. 4 min. 6 min. s.s Figure 16: Transient evolution of Sog (a), Tsg (b), Sog/Tsg (c), Dpp/Scw (d), Sog/Tsg/Dpp/Scw (e), and SBP/BMP (f). 3

31 a. b. Concentration (nm) BMP-Receptor 1 min. 3 min. 4 min. 6 min. s.s. Concentration (nm) Total SBP 1 min. 3 min. 4 min. 6 min. s.s c. d. Concentration (nm) SBP/BMP/Receptor 1 min. 3 min. 4 min. 6 min. s.s. Concentration (nm) Total BMP on Surface 1 min. 3 min. 4 min. 6 min. s.s Figure 17: Transient evolution of BMP-Receptor (a), Total SBP (b), SBP/BMP/Receptor (c), and total BMP on surface (d). 31

32 greater distances [3, ]. However, using PVI staining of Dpp-GFP, it was also shown that over time, BMP is immobilized on the surface and accumulates in a narrow region near the dorsal midline [], presumably as the result of binding to either receptors or another factor such as an SBP. This suggests that the diffusibility of BMP may be a transiently evolving process where, initially the BMP is free to diffuse [3], and later becomes tightly localized in the region of previous signaling []. We find that when equations (1-8) are solved in the absence of receptor interactions with parameter choices over 4 orders of magnitude, many of the requirements for tight localization of BMP are relaxed and the systems exhibits robustness over a wider range of parameters than those found previously [2]. This is in part due to heterodimer formation (Equations 1-3) of the signaling molecules and part due to downstream dimerization reactions (Sog+Tsg and Sog/Tsg + BMP) (our unpublished data and [1]). For Case 3, the upstream robustness gained by the system at steady-state transfers directly the steady-state distributions of occupied receptors. However, for Case 4 it is less clear how the upstream robustness transfers to the level of occupied receptors yet the distributions of BR still exhibit the observed pmad signaling profiles (Figure 3 main text). What is less clear is how patterning is robust with respect to the level of receptors. It is possible that heterodimer formation of receptor subunits may lead to insensitivity by forming heterodimers [1], but positive feedback of an SBP may also compensate for changes in the level of receptors. In the following section, we compare the sensitivity of Cases 1, 2, and 4 to determine the transient robustness for changes in the level of receptors and other parameter values. 7.2 Sensitivity of receptor interactions. Do receptor interactions of Cases 1, 2, and 4 transiently hinder or enhance the robustness of patterning? Also, is the width of pmad signaling resilient to both up-regulation of receptors as previously shown [3, ] and down-regulation of receptors as shown in the zygotic tkv mutant (Figure 18). To help answer these questions we determined the number of cells near the dorsal midline whose level of occupied receptors is greater than a specified threshold 32

33 value for changes in the parameters and levels of the species involved. wt tkv -/- Figure 18: pmad staining in embryos lacking zygotic tkv exhibit wt like signaling profiles. Embryos are homozygous for tkv 7. Choice of parameter values. First, a base case set of parameters for modules I and II was selected and held constant for the different receptor schemes. The parameters used for this study are given in Table 1. Next, parameters were selected for three different versions of module III: Case 1- equilibrium receptor binding, Case 2- receptor binding with endocytosis, or Case 4- positive feedback with endocytosis. We decided to keep the receptor binding and endocytosis rate parameters fixed for the three cases. Since the range of plausible receptor levels spans 4 orders of magnitude (see section ), we vary these for each model, with everything else constant, and choose values so the models are in a similar dynamic range. For Case 1 at 6 minutes, the BMP vs. BMP-receptor curve is linear up to a concentration of 1µM suggesting the transient effects are negligible and we choose a concentration of receptors of 3.2µM and an amplitude of 3nM for BMP-Receptor. To determine the level of receptors for Case 2, we chose the level of receptors that corresponds to the maximum amplitude at 6 min since this would have the sharpest extracellular distribution due to rapid receptor mediated endocytosis. This gives a receptor concentration of 3.2µM. The maximum amplitude for Case 4 gives a receptor concentration of 32 nm which is twice the value used in the main text. Transient results for these choices of parameters are provided in Figures 8, 1, and 13. The BMP-Receptor amplitude at the dorsal midline vs. receptor 33